Derivative is the process of finding rate of change of a quantity with respect to a variable and ‘Anti-derivative’ is the opposite of finding the derivative of a quantity. Finding the anti-derivative of a function is also called as the method of Integration and anti-derivative calculator tool can be used to find the value of it.
Example 1: Find the anti-derivative of the function, f(x) = x5 + 3x-3- 4x2 + 6.
The Power Rule of Integration says that ∫ (x) n dx = x (n+1)/ (n+1) + c
where ‘c’ is a constant
Using the above formula we get,
∫ f(x) dx = x5+1/ (5+1) + 3 * x-3+1/ (-3+1) – 4 * x2+1/ (2+1) + 6x + c
∫ f(x) dx = x6/ 6 + 3 x-2/ (-2) – 4 x3/ 3 + 6x + c
∫ f(x) dx = (1/6) x6 - (3/2) x-2 – (4/3) x3 + 6x + c
Example 2: Find the anti-derivative of the function, f(x) = 1/x.
Derivative and anti-derivative are opposite to each other in the way that if quantity ‘a’ is
the derivative of ‘b’ then the anti-derivative of ‘a’ is the quantity ‘b’.
The derivative of the function, natural logarithm of x written as ln(x) is 1/x.
[dln(x)] = 1/x
Hence the anti-derivative which is the opposite of the derivative tells us that
∫ f(x) dx = ∫ (1/x) dx = ln(x) + c